Let $ \ V \ $ be a vector space and suppose that $ \{U_{n}: n \in \mathbb{N} \}$ are subspaces of $ \ V \ $ . Then prove that , if for every $\ k,\ m\ \in \mathbb{N} $ there exists an $ \ n \in \mathbb{N} \ $ such that $ \ U_{k} \cup U_{m} \subseteq U_{n} $ then $ \ \bigcup_{n=1}^{\infty}U_{n} \ $ is a subspace of $ \ V \ $. $$ $$ I know that union of two subspaces is a subspace iff one is contained in another. I want to use this property but I am not sure how to prove the result ? Any help is appreciating .

Also if $x,y\in \ \bigcup_{n=1}^{\infty}U_{n} \ $, then $x\in U_k$ and $y\in U_m$ for some natural numbers $k,m$ hence they belong to a subspace $U_n$ for some integer $n$ (By the conditions given in the question,) which implies $x+y \in U_n$, so $ x+y \in \ \bigcup_{n=1}^{\infty}U_{n} \ $.